Orthogonal signal generation using vector spreading and combining

ABSTRACT

The present invention provides an approach for quadrature signal generation, which does not require orthogonal reference signals or nearly orthogonal reference signals as an input or given condition. The techniques provided herein can utilize a reference phase shift less than 90° but greater than 0°, along with an inversion to create orthogonal signals. The techniques provided here reduce the number of critical manipulations required from a hardware perspective.

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication No. 60/620,925, titled “Methods and Systems for OrthogonalSignal Generation Using Vector Spreading and Combining,” which was filedon Oct. 22, 2004, and is incorporated herein by reference in itsentirety.

FIELD OF THE INVENTION

The present invention is related to generation of orthogonal signals.

BACKGROUND OF THE INVENTION

Often in signal processing applications, particularly modulation anddemodulation, orthogonal signals are required to transport and recoverlatent information relating to some complex passband waveform.Typically, analog techniques are used in RF modulators and demodulatorsto synthesize a 90° phase shift in a carrier or signal waveform toenable this processing. Passive and active components are used togenerate delays at passband corresponding to phase shift at a frequencyof interest. These techniques possess certain inaccuracies andimplementation tradeoffs. For instance, polyphase filter networkapproaches require multiple refinement stages to obtain reasonableaccuracy while R-C splitters depend on minimizing parasitic elements ofthe interface amplifiers. Quasi-digital approaches are also used forsome applications but trade-off frequency plan, frequency response andpower vs. performance.

What are needed are improved methods and systems for generatingorthogonal signals for quadrature signal generation.

BRIEF SUMMARY OF THE INVENTION

The following summary of the invention provides an understanding of atleast some aspects of the invention. The summary is not an extensiveoverview of the invention. It is not intended to identify key orcritical elements of the invention nor is it intended to delineate thescope of the invention. Its sole purpose is to present some concepts ofthe invention in a simplified form as a prelude to the more detaileddescription that is presented later.

The present invention provides an approach for quadrature signalgeneration, which does not require orthogonal reference signals ornearly orthogonal reference signals as an input or given condition. Thetechniques provided herein can utilize a reference phase shift less than90° but greater than 0°, along with an inversion to create orthogonalsignals. The techniques provided here reduce the number of criticalmanipulations required from a hardware perspective.

Further embodiments, features, and advantages of the present inventions,as well as the structure and operation of the various embodiments of thepresent invention, are described in detail below with reference to theaccompanying drawings. These aspects are indicative of but a few of thevarious ways in which the principles of the invention may be employed,and the present invention is intended to include all such aspects andtheir equivalents. Further features and advantages will be apparent to aperson skilled in the art based on the description set forth hereinand/or may be learned by practice of the invention.

It is to be understood that both the foregoing summary and the followingdetailed description are exemplary and explanatory and are intended toprovide further explanation of embodiments of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES

The accompanying drawings, which are incorporated herein and form partof the specification, illustrate the present invention and, togetherwith the description, further serve to explain the principles of theinvention and to enable a person skilled in the pertinent art to makeand use the invention. In the drawings, like reference numbers indicateidentical or functionally similar elements. Additionally, the leftmostdigit(s) of a reference number identifies the drawing in which thereference number first appears.

FIG. 1 is a vector diagram of the sum of two signals.

FIG. 2 is a vector diagram illustrating that the quadrature component ofa signal can be formed via an inversion of one of the summing vectors.

FIG. 3 is a graphical representation of output vector amplitudevariation versus input amplitude error.

FIG. 4 is a graphical representation of output vector phase variationversus input amplitude error.

FIG. 5 is a graphical representation of phase versus Q.

FIG. 6 is a vector diagram of an imperfect inverse.

FIG. 7 is block diagram of vector signal operations.

FIG. 8 is another block diagram of vector signal operations.

FIG. 9 is another block diagram of vector signal operations.

FIG. 10 is vector diagram illustrating error.

FIG. 11 is vector diagram illustrating distributed error.

FIG. 12 is a generalized vector diagram.

FIG. 13 is a vector diagram of basis vectors.

FIG. 14 is another vector diagram of basis vectors.

FIG. 15 is a block diagram vector quad generator.

FIG. 16 is a circuit diagram of a vector quad generator.

FIG. 17 is a block diagram of a vector quad generator—difference mode.

FIG. 18 is a block diagram of a vector quad generator with transferfunctions.

FIG. 19 illustrates phasor diagrams for vector quad generatorsdifference modes.

FIG. 20 illustrates vector quad generator circuit operation.

FIG. 21 further illustrates vector quad generator circuit operation.

FIG. 22 is a graph of simulated input phase, output amplitude, andoutput phase of a vector quad generator.

FIG. 23 is another graph of simulated input phase, output amplitude, andoutput phase of a vector quad generator.

FIG. 24 is a graph of simulated output amplitude and output phase of avector quad generator.

FIG. 25 is another graph of simulated output amplitude and output phaseof a vector quad generator.

FIG. 26 is a circuit diagram of a vector quad generator and associatedgraph of simulated summer amplitudes, output amplitude, and outputphase.

FIG. 27 is another circuit diagram of a vector quad generator andassociated graph of simulated summer amplitudes, output amplitude, andoutput phase.

FIG. 28 is a circuit diagram of a vector quad generator difference modecircuit.

FIG. 29 is a circuit diagram of a vector quad generator difference modecircuit with a difference amplifier.

FIG. 30 is a block diagram/circuit diagram of a vector quad generatorwithout subtraction.

FIG. 31 is a block diagram of a vector quad generator, summer approach.

FIG. 32 is another block diagram of a vector quad generator withtransfer functions.

FIG. 33 illustrates additional phasor diagrams for a vector quadgenerator.

FIG. 34 is another circuit diagram of a vector quad generator.

FIG. 35 is another circuit diagram of a vector quad generator.

FIG. 36 is a simulated graphical illustration of quadrature phase versusamplitude mismatch.

FIG. 37 is a simulated graphical illustration of quadrature phase versusinput frequency.

FIG. 38 is a simulated graphical illustration of quadrature phase versusinput phase.

FIG. 39 is a simulated graphical illustration of quadrature phase versus0/180 input phase error.

FIG. 40 is a simulated graphical illustration of quadrature phase versus0/180 input amplitude error.

FIG. 41 is a simulated graphical illustration of a current mode summer.

FIG. 42 is a simulated graphical illustration of quadrature phase andamplitude mismatch.

FIG. 43 is a Monte Carlo simulated graphical illustration of quadraturephase and amplitude mismatch.

FIG. 44 is an un-optimized Monte Carlo simulated graphical illustrationof quadrature phase and amplitude mismatch.

FIG. 45 is a block diagram of a tuned differential R-C quad vectorgenerator.

FIG. 46 is a block diagram of a tuned differential R-C quad vectorgenerator with transfer functions.

FIG. 47 illustrates phasor diagrams for a tuned differential R-C quadvector generator.

FIG. 48 is a circuit diagram of a tuned differential R-C quad vectorgenerator, differential cascade implementation.

FIG. 49 is a simulated graphical illustration of amplitude and phaseversus input amplitude mismatch for a tuned differential R-C quad vectorgenerator.

FIG. 50 is a simulated graphical illustration of quadrature phase versusfrequency for a tuned differential R-C quad vector generator.

FIG. 51 is a simulated graphical illustration of quadrature phase versusphase mismatch for a tuned differential R-C quad vector generator.

FIG. 52 is a Monte Carlo simulated graphical illustration for a tuneddifferential R-C quad vector generator.

FIG. 53 is an un-optimized Monte Carlo simulated graphical illustrationfor a tuned differential R-C quad vector generator.

FIG. 54 is a simulated graphical illustration of amplitude limited bytail current and loaded Q, for a tuned differential R-C quad vectorgenerator.

DETAILED DESCRIPTION OF THE INVENTION

1. Introduction

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which embodiments of theinvention are illustrated. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein. Rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the invention to those skilled in the art.

It will be understood that when an element is referred to as being“connected” or “coupled” to another element, it can be directlyconnected or coupled to the other element or intervening elements may bepresent. In contrast, when an element is referred to as being “directlyconnected” or “directly coupled” to another element, there are nointervening elements present. Moreover, each embodiment described andillustrated herein includes its complementary conductivity typeembodiment as well.

In the following detailed description, numerous specific details are setforth in order to provide a thorough understanding of the invention.However, it will be understood by those skilled in the art that thepresent invention may be practiced without these specific details. Inother instances, well-known methods, procedures, components and circuitshave not been described in detail so as not to obscure the presentinvention.

The present invention is directed to methods and systems for orthogonalsignal generation and quadrature signal generation, which does notrequire orthogonal reference signals or nearly orthogonal referencesignals as an input or given condition. The technique can utilize areference phase shift less than 90° but greater than 0° along with aninversion to create orthogonal signals. This minimizes the number ofcritical manipulations required from a hardware perspective.

In a more general context this technique may extend to encompass linearoperations involving 2 basis vectors and their inverses. A minimal basisset consists of 3 of these vectors. An infinite number can be used. Thebasis vectors are not required to be orthogonal but possess significantorthogonal components to develop the desired coordinate transformationswhile minimizing errors.

The simplest approaches will always involve only 2 parallel calculationsgiven the basis set and at least one inverse (hereafter extended minimalbasis set).

In Sections 1 through 4 a particularly simple example is presented withthe associated trigonometric identities. In Section 5 a general vectorrepresentation is developed. Section 6 provides some implementationstrategies, with performance analysis.

Throughout this description, complex arithmetic will be utilized. Sincethe signals of interest lie in the complex plane the vectors can berepresented by complex numbers arranged in the Cartesian coordinatesystem. Occasionally magnitude and angle of the vectors (signals) willalso be represented.

2. I Vector

Suppose two passband signals are summed, possessing the same envelopemagnitudes yet different phases. For convenience in representation andvisualization, assume that the signals are carriers without modulation.Although, this is not a significant restriction and can be waived forvery practical scenarios involving approximately narrowband passbandsignals employing PM, AM and FM. The approximation used here assumesthat f_(c)>>B where,

f_(c) Δ Carrier Passband Center Frequency

BΔ Effective Bandwidth of the Modulation of Interest

Proceeding, we find the sum of the two passband signals as;R(t)sin(ω_(c) t+β(t))=A(t)(sin ω_(c) t+γ(t))+A(t)sin(ω_(c) t+γ(t)+φ)

R(t)Δ Resulting Combined Amplitude Function

β(t)Δ 0 Resulting Phase of Combined Passband Signal

A(t)Δ Amplitude Function Describing the Passband Signals

γ(t)Δ Phase Function Describing the Passband Signals

φΔ Phase Shift between two Passband Carriers 0°<φ≦90°

If the summed carriers are not modulated then A(t)=k_(A) (a constant)and γ(t)=k_(γ) another constant. However, it can be shown that providedthe amplitude function changes only slightly and/or the phase functionchanges slightly over the interval associated with the time required toimplement φ, A(t), and γ(t) can be regarded as virtually constant forthe purpose of the following analysis so that;R sin(ω_(c)t+β)≅A sin ω_(c)t+A sin(ω_(c)t+φ)

Reducing this result using analytic geometry and algebra, yields;$R = {2A\quad\cos\frac{\phi}{2}}$$\phi = {{arcos}\left( {\frac{R^{2}}{2A} - 1} \right)}$$\beta = {{\arctan\left( \frac{R^{2}}{2\sqrt{1 - \phi^{2}}} \right)} = {{{\tan^{- 1}\left( \frac{1 + {\cos\quad\phi}}{\sin\quad\phi} \right)}\quad\ldots\quad A} = 1}}$

Referring to FIG. 1, consider the following vector diagram of the sum oftwo such signals.

Notice that the angular distance between the vectors is φ. If the twovectors possess equal amplitudes, A, then they sum to a new vector ofamplitude R at an angle of 0°. If the amplitudes of the two vectors aredifferent then they sum to an angle other than 0°. For purposes of thispaper analysis it is assumed that 0° is the in phase or I signal while90° is the Q signal component. Then I is simply formed from this sum.

3. Q Vector Generation

The quadrature component of the signal can be formed via an inversion ofone of the summing vectors illustrated in the previous section and theremaining vector as illustrated in FIG. 2.

Notice that {right arrow over (V)}₁+{right arrow over (V)}₄ form the Ireference as illustrated in the previous section. By combing (throughvector summation) −{right arrow over (V)}₁+{right arrow over (V)}₄ anorthogonal vector can be formed also. This can be represented by thefollowing simple math.−{right arrow over (V)} ₁ +{right arrow over (V)} ₄ =R ₉₀ sin(ω_(c)t+(β−π/4))=−A sin(ω_(c) t+φ/2)+A sin(ω_(c) t−φ/2)=−A sin(ω_(c)t+(φ/2−π/2))+A sin(ω_(c) t−φ/2)

Similarly,{right arrow over (V)} ₁ −{right arrow over (V)} ₄ =A sin(ω_(c) t+φ/2)−Asin(ω_(c) t−φ/2)=2A sin(φ/2)cos(ω_(c) t)

The equivalence of these vector operations is apparent. Also it isapparent that the I vector is orthogonal to Q as shown in previouscalculations. The results are repeated here side by side for comparison;{right arrow over (I)}={right arrow over (V)} ₁ +{right arrow over (V)}₄=2A cos(φ/2)sin(ω_(c) t){right arrow over (Q)}={right arrow over (V)} ₁ −{right arrow over (V)}₄=2A sin(φ/2)cos(ω_(c) t)

The first term in each equation is a scalar for the vector, dependent onthe angle φ/2. The second term in each equation in fact is the phasor ofinterest for I and Q components. Notice that the resulting amplitudesfor the orthogonal signals are only equivalent if φ/2=45°. This isobviously a preferred angle but is not required since the output ({rightarrow over (I)},{right arrow over (Q)}) vectors are always orthogonalregardless of the scalar component of the vectors.

4. Error Anomalies

One advantage of generating the orthogonal vectors in this manner isinvestment in hardware, power, complexity, etc., given a desiredprecision. In other words, other techniques typically require more poweror more hardware for a given accuracy. Nevertheless, this approach hassome error associated with the parameters of the generating vectors andassociated hardware which should be examined.

The most obvious error is associated with amplitude imbalance of thegenerating vectors, say {right arrow over (V)}₁ and {right arrow over(V)}₄.

Consider the following;R sin(ω_(c) t+β)=A ₁ sin ω_(c) t+A ₂ sin(ω_(c) tφ)=A sin ω_(c) t+ΔAsin(ω_(c) t+φ)where Δ represents a Multiplicative Error Term and where Δ=1 Correspondsto No Error

This can be written as;R sin(ω_(c) t+β)=A ₁ sin ω_(c) t+A ₂(cos φ sin(ω_(c) t)+sin φ cos(ω_(c)t))=(AΔA)sin(ω_(c) t)+ΔA sin φ cos(ω_(c) t)

When Δ>1 the resulting vector rotates counter clockwise from 0° and whenΔ<1 it rotates clockwise. Of course, one may argue that this rotation of{right arrow over (V)}₁+{right arrow over (V)}₄, is after all arbitrarysince the reference plane of I can be rotated and that a variation inφ/2 should therefore not affect our interpretation of 0°. This is true.However, the difficulty arises when the orthogonal vector is producedrelative to the new I reference if Δ still exists when creating theinversion.

It is convenient to refer to equations based on φ/2 for furtheranalysis.${{\overset{\rightarrow}{V}}_{1} + {\overset{\rightarrow}{V}}_{4}} = {{A\left( {{\sin\left( {{\omega_{c}t} + \frac{\phi}{2}} \right)} + {\sin\left( {{\omega_{c}t} - \frac{\phi}{2}} \right)}} \right)} + \underset{{error}\quad{term}}{\underset{︸}{{A\left( {\Delta - 1} \right)}{\sin\left( {{\omega_{c}t} - \frac{\phi}{2}} \right)}}}}$

In this example all of the error term was allocated to {right arrow over(V)}₄ but could have been assigned (in part or whole) to {right arrowover (V)}₁ as well. Notice that the affect is to pull the desired 0° Ireference slightly from 0° depending on the value of Δ. Of course this0° reference is arbitrary and the error angle may be redefined as 0°even if this calculation is not 0° due to a value of Δ other than 1.However, the formation of the quadrature vector follows as;{right arrow over (V)} ₁ −{right arrow over (V)} ₄ =A(sin(ω_(c)tφ/2)−sin(ω_(c) t−φ/2))−A(Δ−1)sin(ω_(c) t−φ/2)

From the above equations it is easy to deduce the absolute phase errorsinvolved as;${I\quad{vector}\quad{error}\quad\underset{\_}{\Delta}\quad I_{\phi_{ɛ}}} = {\tan^{- 1}\left( \frac{{A\left( {\Delta - 1} \right)}{\sin\left( \frac{- \phi}{2} \right)}}{{{A\left( {\Delta - 1} \right)}{\cos\left( \frac{- \phi}{2} \right)}} + {\sqrt{2} \cdot A}} \right)}$${Q\quad{vector}\quad{error}\quad\underset{\_}{\Delta}\quad Q_{\phi_{ɛ}}} = {{\tan^{- 1}\left( \frac{{A\left( {\Delta - 1} \right)}{\cos\left( \frac{- \phi}{2} \right)}}{{{- {A\left( {\Delta - 1} \right)}}{\sin\left( \frac{- \phi}{2} \right)}} + {\sqrt{2} \cdot A}} \right)} - {90{^\circ}}}$

The −90° in the second term of the Q_(φ) _(E) is included to registerthe error relative to the imaginary axis.

These errors (I_(φ) _(E) , Q_(φ) _(E) ) are cumulative. That is, theyalways add to increase phase error in one direction or the other,relative to 90° for this example. Minimizing one error also minimizesthe other error from the perspective of this specific mechanism of errorgeneration. Since the input parameter A is varied to produce the phaseerror this anomaly is labeled AM-PM conversion.

4.1. Output Amplitude Imbalance

Previous analysis has illustrated that phase errors at the output of theproposed I, Q generator are produced by (relative) amplitude errors ofthe input vectors. This may be considered as AM to PM conversion.However, the reverse situation exists also.

Whenever, the generating vectors are not ±φ/2=±45°, the output vectorsare amplitude imbalanced. This is PM to AM conversion.

The equations;{right arrow over (I)}=2A cos(φ/2)sin(ω_(c) t){right arrow over (Q)}=2A sin(φ/2)cos(ω_(c) t)

illustrate that {right arrow over (I)} and {right arrow over (Q)} areonly equivalent in amplitude if ±φ/2=±45°. This is therefore a preferredangle, although not strictly required for practical implementation.Whenever φ/2 is such that I decreases then Q increases proportionallysince the scalar error follows sin and cos functions.

This error is usually modest for any reasonable implementation and canbe reduced further by non-linear operations. It is essential however,that the non-linear operation minimize AM to PM conversion since phaseaccuracy is the goal.

4.2. Error Sensitivities

The first graph illustrates that the vector amplitude resulting from{right arrow over (V)}₁+{right arrow over (V)}₂ will vary if |{rightarrow over (V)}₂| varies. The input error is swept from 0 to 2 for theerror multiplier ΔA. FIG. 3 is a graph of I Output Vector AmplitudeVariation vs. Input Amplitude Error for V2 (ΔA).

Notice that for ΔA=1 the output vector amplitude is √{square root over(2)}.

More importantly, the phase of this vector sum also is modifiedaccording to the graph FIG. 4, which illustrates I Output Vector PhaseVariation vs. Input Amplitude Error for V2 (ΔA) AM to PM Conversion.Fortunately the sensitivity about 0° is fairly low, on the order of, 4°for a 10% variance of ΔA.

FIG. 5 is a graph of I Phase vs. Q for a Linear Sweep of ΔA between 0and 2. FIG. 5 shows that when I varies so does Q in such a manner as todouble the variance from orthogonal.

Note that when I=0°, Q=−90°. Also, it is important to note the linearrelationship between the two.

4.3. Error Due To Imperfect Inverse

In the example presented thus far the extended minimal basis setconsists of {right arrow over (V)}₁, {right arrow over (V)}₄ and −{rightarrow over (V)}₄. If −{right arrow over (V)}₄ is imperfect then furthererror will result unless it is cancelled. In the simple case presented,if |{right arrow over (V)}₄|=|{right arrow over (V)}₁| then an error inphase Θ_(e) from 180° for −{right arrow over (V)}₄ (compared to {rightarrow over (V)}₄) will translate to an error in orthogonality of Θ_(e)/2for {right arrow over (V)}₁−{right arrow over (V)}₄. Consider FIG. 6.

4.4. Obtaining the Perfect Inverse

In Section 4.3 the error due to an imperfect inverse is found to beΘ_(e)/2. Therefore it is desirable to minimize this error. This can beaccomplished by reversing the roles of {right arrow over (V)}₁ and{right arrow over (V)}₄ after the fact and completing the linearoperations. Another way to view this is to introduce −{right arrow over(V)}₁ by a virtual or differential operation. For purposes of this papersuch a virtual operation is considered as a subtraction at a summingnode rather than an inversion. Although the operations aremathematically equivalent they possess different physical significancewhen related to hardware implementation. Consider two diagrams in FIG.7.

If G is real and G=1 then both diagrams are equivalent. However, if G≠1or if G is complex then the diagrams are not equivalent. For this reasondifferential subtraction nodes are preferred to avoid the errorintroduced by {tilde over (G)}. The errors presented in Section 4.3 areerrors related to {tilde over (G)}.

Consider FIG. 8, which creates a more robust inversion under certaincircumstances.

Therefore, if difference operations are preferred and considered idealor nearly ideal, as compared to inversion plus summing, then the blockdiagram in FIG. 8 generates the inverted signals with minimum error,provided the two difference nodes are matched in every way.

Now consider the vector signal operations of FIG. 9, where:{right arrow over (V)} ₀ ={right arrow over (V)} _(A) −{right arrow over(V)} _(C) 0° degree vector output{right arrow over (V)} ₉₀ ={right arrow over (V)} _(B) −{right arrowover (V)} _(C) 90° degree vector output∴jV _(A) −j{right arrow over (V)} _(C) ={right arrow over (V)} _(B)−{right arrow over (V)} _(C) (required for orthogonal outputs)Then,${\overset{\rightharpoonup}{V}}_{C} = {\frac{{\overset{\rightharpoonup}{V}}_{A} + {\overset{\rightharpoonup}{V}}_{B}}{2} - {j\frac{\left( {{\overset{\rightharpoonup}{V}}_{A} - {\overset{\rightharpoonup}{V}}_{B}} \right)}{2}}}$

The rotation by j as well as the scaling, summation, subtraction, etc.,are considered to be linear operators. Hence there are many linearcircuit options which can produce these vectors. An alternate form isvaluable as well.$V_{C} = {{\frac{{\overset{\rightharpoonup}{V}}_{A}}{2}\left( {1 - j} \right)} + {\frac{{\overset{\rightharpoonup}{V}}_{B}}{2}\left( {1 + j} \right)}}$

Whenever {right arrow over (V)}_(B)=−{right arrow over (V)}_(A) then${\overset{\rightharpoonup}{V}}_{C} = \frac{{jV}_{A}}{2}$

Now suppose that {right arrow over (V)}_(A)≠{right arrow over (V)}_(B)due to some error. Then{right arrow over (V)} _(A) ={right arrow over (V)} _(B)+{right arrowover (Δ)}_(AB)Where {right arrow over (Δ)}_(AB) is required to correct {right arrowover (V)}_(B) so that V_(A) and V_(B) are antipodal and |{right arrowover (V)}_(A)|=|{right arrow over (V)}_(B)+{right arrow over (Δ)}_(AB)|.FIG. 10 illustrates this with a simplified vector diagram

Furthermore, suppose that the error angle and amplitude error can bedistributed equally between V_(A) and V_(B) such that the equivalentdiagram is drawn in FIG. 11.

Notice that the total error angle Θ_(e) is preserved and the anglebetween {right arrow over (V)}_(A) and {right arrow over (V)}_(B) isunchanged. Furthermore, the magnitudes of {right arrow over (V)}_(A) and{right arrow over (V)}_(B) are unchanged.

The new vectors {right arrow over (V)}_(Aε) and {right arrow over(V)}_(Bε) are defined as${\overset{\rightharpoonup}{V}}_{A\quad ɛ} = {{\overset{\rightharpoonup}{V}}_{A} + \frac{{\overset{\rightharpoonup}{\Delta}}_{AB}}{2}}$${\overset{\rightharpoonup}{V}}_{B\quad ɛ} = {{\overset{\rightharpoonup}{V}}_{B} + \frac{{\overset{\rightharpoonup}{\Delta}}_{AB}}{2}}$

Then {right arrow over (V)}₀ is found from;${\overset{\rightharpoonup}{V}}_{0} = {{{\overset{\rightharpoonup}{V}}_{A\quad ɛ} - {\overset{\rightharpoonup}{V}}_{C}} = {{\frac{{\overset{\rightharpoonup}{V}}_{A\quad ɛ} - {\overset{\rightharpoonup}{V}}_{B\quad ɛ}}{2}\left( {1 + j} \right)} = {\frac{{\overset{\rightharpoonup}{V}}_{A} - {\overset{\rightharpoonup}{V}}_{B}}{2}\left( {1 + j} \right)\underset{\_}{{\therefore{V_{A_{ɛ}} - V_{B_{ɛ}}}} = {V_{A} - V_{B}}}}}}$Also, V₉₀ is found in similar fashion;${\overset{\rightharpoonup}{V}}_{90} = {{{\overset{\rightharpoonup}{V}}_{B\quad ɛ} - {\overset{\rightharpoonup}{V}}_{C}} = {\frac{{\overset{\rightharpoonup}{V}}_{A} - {\overset{\rightharpoonup}{V}}_{B}}{2}\left( {{- 1} + j} \right)}}$

It should be obvious by inspection that V₀ and V₉₀ are orthogonal.

Notice that the errors $\frac{\Delta_{AB}}{2}$cancel entirely. This occurs due to the use of differential architectureand differential summing nodes. This permits the errors Δ_(AB)/2 toexist simultaneously at the inputs V_(Aε) and V_(Bε) in exactly the sameamplitude and phase. Therefore, the errors appear as common mode andcompletely cancel at the output provided that {right arrow over (V)}_(C)is given as specified.5. General Presentation

Sections 2 through 3 presented a relatively simple example whichutilized the basic principle of manipulating 2 vector basis functionsand at least one of their inverses to obtain orthogonal vectors.Beginning with the extended basis function set a single parallel stageof linear math operations can be applied to obtain an orthogonal result.A minimum of two linear operations are required.

In order to generalize the presentation the following vectorrepresentation is adopted, with reference to FIG. 12.

Four vectors are constructed to encompass possible extended basis setsfrom all 4 quadrants for this presentation. A minimal extended basis set(excluding degenerate cases) consists of 3 vectors. It should be evidentthat the basis set could be composed of an infinite numbers of vectors.They are not required to be orthogonal and their respective magnitudesare not required to be equal. Hence, the following basic equations arerecorded using complex math notation.{tilde over (V)} ₁=α₁ +jβ ₁{tilde over (V)} ₂=α₂ +jβ ₂{tilde over (V)} ₃=α₃ +jβ ₃{tilde over (V)} ₄=α₄ +jβ ₄α, β may be positive or negative numbers.

In this representation ˜ denotes a complex number and the subscriptsdenote coordinate quadrant. Without loss of generality {tilde over (V)}₁is established as a reference vector with φ/2=45° or π/4 radians. With{tilde over (V)}₁ as a reference the vector cross errors are defined asthe incremental change required to orthogonalize the pair wise vectors,while maintaining the desired normalized vector magnitude (i.e., allphase and amplitude errors removed). Hence,${{\alpha_{1} + {j\quad\beta_{1}}}} = {{\alpha_{2} + {j\quad\beta_{2}} + {\overset{\sim}{\Delta}}_{12}}}$$\frac{\alpha_{1}}{\beta_{1}} = {{\frac{- \left( {\alpha_{2} + {R_{e}{\overset{\sim}{\Delta}}_{12}}} \right)}{\left( {\beta_{2} + {I_{m}{\overset{\sim}{\Delta}}_{12}}} \right)}\therefore\alpha_{1}} = {- \left( {\alpha_{2} + {R_{e}{\overset{\sim}{\Delta}}_{12}}} \right)}}$$\beta_{1} = \left( {\beta_{2} + {I_{m}{\overset{\sim}{\Delta}}_{12}}} \right)$${\alpha_{1} = {- \alpha_{2}}},{{R_{e}{\overset{\sim}{\Delta}}_{12}} = 0}$${\beta_{1} = \beta_{2}},{{I_{m}{\overset{\sim}{\Delta}}_{12}} = 0}$α₁ = α₂ = β₁ = β₂for zero error case

Forming all possible cross errors yields the following cross errormatrix.$\left( \overset{\sim}{\Delta} \right){\underset{\_}{\Delta}\begin{pmatrix}0 & {\overset{\sim}{\Delta}}_{12} & {\overset{\sim}{\Delta}}_{13} & {\overset{\sim}{\Delta}}_{14} \\{- {\overset{\sim}{\Delta}}_{12}} & 0 & {\overset{\sim}{\Delta}}_{23} & {\overset{\sim}{\Delta}}_{24} \\{- {\overset{\sim}{\Delta}}_{13}} & {- {\overset{\sim}{\Delta}}_{23}} & 0 & {\overset{\sim}{\Delta}}_{34} \\{- {\overset{\sim}{\Delta}}_{14}} & {- {\overset{\sim}{\Delta}}_{24}} & {- {\overset{\sim}{\Delta}}_{34}} & 0\end{pmatrix}}$Where;{tilde over (Δ)}₁₂=−{tilde over (Δ)}₂₁{tilde over (Δ)}₁₃=−{tilde over (Δ)}₃₁{tilde over (Δ)}₁₄=−{tilde over (Δ)}₄₁{tilde over (Δ)}₂₃=−{tilde over (Δ)}₃₂{tilde over (Δ)}₂₄=−{tilde over (Δ)}₄₂{tilde over (Δ)}₃₄=−{tilde over (Δ)}₄₃Also the dot and cross products will be needed. For instance;$\begin{matrix}{{\left( \quad \right.\overset{{\overset{\sim}{V}}_{1}}{\overset{︷}{\alpha_{1} + {j\quad\beta_{1}}}}{\left. \quad \right) \cdot \left( {\left( \overset{{\overset{\sim}{V}}_{2}}{\overset{︷}{\alpha_{2} + {j\quad\beta_{2}}}} \right) + {\overset{\sim}{\Delta}}_{12}} \right)}} = 0} \\{= {{{\alpha_{1} + {j\quad\beta_{1}}}}{{\alpha_{2} + {j\quad\beta_{2}} + {\overset{\sim}{\Delta}}_{12}}}\cos\quad\Theta_{12}}}\end{matrix}$${\left( {\alpha_{1} + {j\quad\beta_{1}}} \right) \times \left( {\alpha_{2} + {j\quad\beta_{2}} + {\overset{\sim}{\Delta}}_{12}} \right)} = {{{\alpha_{1} + {j\quad\beta_{1}}}}{{\alpha_{2} + {j\quad\beta_{2}} + {\overset{\sim}{\Delta}}_{12}}}\sin\quad{\Theta_{12} \cdot \overset{\_}{n}}}$

Whenever the dot product is zero then the vectors are orthogonal.Whenever the vectors are orthogonal then the magnitude of the crossproduct is the product of the individual magnitudes and the product ismaximized.

These identities and vector relationships will be utilized heavily inthe following treatment.

5.1. Particular Case 1

Suppose the example analyzed in Sections 2 through 3 is presented withthe alternate representation using vector notation and complex numbers.Then the following operations are readily identified.$\left. \begin{matrix}\begin{matrix}{{\overset{\sim}{V}}_{1} = {\alpha_{1} + {j\quad\beta_{1}}}} \\{{\overset{\sim}{V}}_{4} = {\alpha_{4} + {j\quad\beta_{4}}}}\end{matrix} \\{{- {\overset{\sim}{V}}_{1}} = {{{- \alpha_{1}} - {j\quad\beta_{1}}} = {\alpha_{3} + {j\quad\beta_{3}}}}}\end{matrix} \right\}\quad{extended}\quad{minimal}\quad{basis}\quad{set}$The formation of orthogonal vectors proceeds{tilde over (V)} ₀ ={tilde over (V)} ₁ +{tilde over (V)} ₄+{tilde over(Δ)}₁₄{tilde over (V)} ⁻⁹⁰ =−{tilde over (V)} ₁ +{tilde over (V)} ₄+{tildeover (Δ)}₁₄

The dot product of these two vectors is identically zero only whenΔ₁₄=0. The dot product and cross product errors of the original vectorsums is found from;$\frac{{Dot}\quad{Product}\quad{Error}}{\begin{matrix}{{\therefore{{\overset{\sim}{V}}_{0} \cdot {\overset{\sim}{V}}_{- 90}}} = {{{\overset{\sim}{V}}_{4}}^{2} - {{\overset{\sim}{V}}_{1}}^{2} + {{\overset{\sim}{\Delta}}_{14}}^{2} + {2{{\overset{\sim}{V}}_{4}}{{\overset{\sim}{V}}_{1}}\cos\quad\Theta_{14}} +}} \\{{2{{\overset{\sim}{V}}_{1}}{{\overset{\sim}{\Delta}}_{14}}{\cos\left( \Theta_{1{\Delta 14}} \right)}} + {2{{\overset{\sim}{V}}_{4}}{{\overset{\sim}{\Delta}}_{14}}{\cos\left( \Theta_{4{\Delta 14}} \right)}}}\end{matrix}}$

It is useful to examine some properties of this particular equationunder various circumstances. $\begin{matrix}{{{If}\quad{\overset{\sim}{V}}_{4}} = {{{\overset{\sim}{V}}_{1}^{*}\quad{then}\quad{V_{0} \cdot V_{- 90}}} = 0}} & {{Observation}\quad 1} \\{{{If}\quad{\overset{\sim}{\Delta}}_{14}} = {{0\quad{then}\quad{V_{0} \cdot V_{- 90}}} = 0}} & {{Observation}\quad 2} \\{{{{When}\quad\cos\quad\Theta_{1{\Delta 14}}} = {{0\quad{then}\quad{V_{0} \cdot V_{- 90}}} = {0\quad{and}}}}{{vice}\quad{versa}}} & {{Observation}\quad 3} \\{{{{If}\quad{\overset{\sim}{\Delta}}_{14}} = {{\overset{\sim}{V}}_{1} = {\overset{\sim}{V}}_{4}^{*}}}{{{\overset{\sim}{\Delta}}_{14}}^{2} = {\left( {{\alpha_{1} - \alpha_{14}}} \right)^{2} + \left( {{\beta_{1} - \beta_{4}}} \right)^{2}}}} & {{Observation}\quad 4} \\{{{\Theta_{\Delta 14} = {\arctan\left( \frac{\beta_{1} + \beta_{4}}{\alpha_{1} - \alpha_{4}} \right)}};{{\alpha_{1} - \alpha_{4}} > 0};}{1^{st},{4^{th}\quad{quadrants}}}{{\Theta_{\Delta 14} = {{\arctan\left( \frac{\beta_{1} + \beta_{4}}{\alpha_{1} - \alpha_{4}} \right)} + {180{^\circ}}}};{{\alpha_{1} - \alpha_{4}} > 0};}{2^{nd},{3^{r\quad d}\quad{quadrants}}}\begin{matrix}{\Theta_{1\quad{\Delta 14}} = {\Theta_{14} + \Theta_{4\quad{\Delta 14}}}} & {{Angle}\quad{between}\quad{\overset{\sim}{V}}_{1}\quad{and}\quad{the}\quad{error}\quad{vector}\quad{\overset{\sim}{\Delta}}_{14}} \\{\Theta_{\Delta 14} = {\Theta_{4} + \Theta_{4\quad{\Delta 14}}}} & {{Absolute}\quad{error}\quad{vector}\quad{angle}\quad{for}\quad{\overset{\sim}{\Delta}}_{14}} \\{\Theta_{14} = {{{- 45}{^\circ}} + \Theta_{4}}} & {{Difference}\quad{angle}\quad{between}\quad{\overset{\sim}{V}}_{1}\quad{and}\quad{\overset{\sim}{V}}_{4}}\end{matrix}\begin{matrix}\begin{matrix}{{Cross}\quad{Product}} \\{\quad{{{\quad\overset{\quad}{\overset{\sim}{V}}}_{0} \times {\overset{\quad}{\overset{\sim}{V}}}_{- 90}}\quad = \quad{{{\text{(}{\overset{\quad}{\overset{\sim}{V}}}_{1}} + {\overset{\quad}{\overset{\sim}{V}}}_{4} + {{\overset{\quad}{\overset{\sim}{\Delta}}}_{14}\text{)} \times \text{(}} - V_{1} + V_{\quad 4} - {\Delta_{14}\text{)}}}\quad =}}\quad}\end{matrix} \\{{2\quad\text{(}\text{}{\overset{\quad}{\overset{\sim}{V}}}_{1}\text{}\quad\text{}{\overset{\quad}{\overset{\sim}{V}}}_{4}\text{}\quad{\sin\left( \Theta_{14} \right)}} + {{\Delta_{14}}\quad{V_{\quad 4}}\quad{\sin\left( \Theta_{4\quad{\Delta 14}} \right)}} + {\text{}{\overset{\quad}{\Delta}}_{14}\text{}\quad\text{}V_{1}\text{}\quad{\sin\left( \Theta_{\quad{1{\Delta 14}}} \right)}\text{)}\quad\overset{\quad\_}{\quad n}}}\end{matrix}} & {{Observation}\quad 5}\end{matrix}$If the error |Δ₁₄| tends to zero then {tilde over (V)}₀×{tilde over(V)}⁻⁹⁰=|{tilde over (V)}₁||{tilde over (V)}₄|·{right arrow over (n)}

Both the dot and cross products are excellent metrics for measuringorthogonality and provide a means for estimating sensitivity for thevarious algorithms.

5.2. Case 2

Using the previous notation two vectors are formed as follows;V ₀ ={tilde over (V)} ₁−({tilde over (V)} ₃+{tilde over (Δ)}₁₄)V ₊₉₀ ={tilde over (V)} ₂−({tilde over (V)} ₄+{tilde over (Δ)}₂₄)

The basis vectors are redrawn in FIGS. 13 and 14 for reference.{tilde over (V)} ₄₅ ={tilde over (V)} ₁−({tilde over (V)} ₃+{tilde over(Δ)}₁₃)=(α₁−α₃)+(β₁−β₃)j−{tilde over (Δ)} ₁₃{tilde over (V)} ₁₃₅=({tilde over (V)} ₂+2{tilde over (Δ)}₁₂)−({tildeover (V)} ₄+{tilde over (Δ)}₂₄)=(α₂−α₄)+(β₂−β₄)j−{tilde over (Δ)}₂₄+2{tilde over (Δ)}₁₂α₁₃=α₁−α₃β₁₃=β₁−β₃α₂₄=α₂−α₄β₂₄=β₂−β₄

All of the correction (error) vectors are included such that V₄₅ andV₁₃₅ are orthogonal. Notice the vectors on the real axis and imaginaryaxis. V₁₃₅ and V₄₅ are composed from those vectors, {right arrow over(x)}, −{right arrow over (x)}, {right arrow over (y)}. This is exactlythe same form as that given in Section 5.1 for Case 1, i.e.,{right arrow over (V)} ₁₃₅ =−{right arrow over (x)}+{right arrow over(y)}V ₄₅ ={right arrow over (x)}+{right arrow over (y)}

This is the same form as;{right arrow over (V)} ₁ +{right arrow over (V)} ₄ ={right arrow over(V)} ₀{right arrow over (V)} ₁ −{right arrow over (V)} ₄ ={right arrow over(V)} ₉₀

All cases regardless of how many original basis vectors will reduce tothe trivial minimal basis set of 3 vectors.

6. Implementation Strategies

A block diagram for the specific example presented in Sections 2 through3 is illustrated FIG. 15.

Notice that the outputs are formed from the original extended basisvectors {right arrow over (V)}₁, −{right arrow over (V)}₁, {right arrowover (V)}₂ where {right arrow over (V)}₂ is quasi orthogonal to {rightarrow over (V)}₁, −{right arrow over (V)}. That is, {right arrow over(V)}₂ possesses a significant orthogonal component even though someerror is present as well. All versions may be shown to reduce to thisbasic form, under linear operation and coordinate transformation.

6.1. First Implementation

Given the block diagram above a circuit may be constructed according tothe schematics described below. First some ideal block diagrams areprovided with some equations which explain the vector operations inpolar form. Finally a circuit is given which details an implementationwith actual components.

FIG. 16 illustrates a Vector Quad Gen Block Diagram/ABM SimulationCircuit.

FIG. 17 illustrates a Vector Quad Gen Simplified BlockDiagram—Difference Approach.

FIG. 18 illustrates Vector Quad Gen Block Diagram/Transfer Functions.

FIG. 19 illustrates Vector Quad Gen Difference Mode Phasor Diagrams.With an R-C implementation of the 90° phase shift network, gain G1 andphase error φ_(E1) become a function of the input phase mismatch,φ_(E2). This dependency results in a cancellation of output quadraturephase error in response to input phase error (φ_(E1)).

FIG. 20 illustrates Vector Quad Gen Circuit Operation. The performanceof the architecture is evaluated through simulations with idealcomponents. The transmission line characteristics are altered tosimulate input phase mismatch.

FIG. 21 further illustrates Vector Quad Gen Circuit Operation. Thisarchitecture demonstrates excellent output amplitude matching and zeroquadrature phase error in response to varying input phase.

FIG. 22 graphically illustrates Vector Quad Gen Ideal CircuitSimulation.

FIG. 23 further graphically illustrates Vector Quad Gen Ideal CircuitSimulation. The circuit provides excellent output amplitude matching andzero quadrature phase error in response to varying input amplitudemismatch.

FIG. 24 further illustrates Vector Quad Gen Ideal Circuit Simulation.Frequency variations produce output amplitude mismatch and zero phaseerror.

FIG. 25 further illustrates Vector Quad Gen Ideal Circuit Simulation.R-C phase splitter variations (±10% shown) produce output amplitudemismatch and zero phase error.

FIG. 26 further illustrates Vector Quad Gen Ideal Circuit Simulation.FIG. 26 illustrates high sensitivity to differencer common nodeamplitude: 5% amplitude variation results in approximately 4° ofquadrature phase error.

FIG. 27 further illustrates Vector Quad Gen Ideal Circuit Simulation. 5%amplitude variation at differencer plus input node results inapproximately 2° of quadrature phase error.

FIG. 28 illustrates Vector quad Gen Difference Mode CircuitImplementation Circuit 1.

FIG. 29 illustrates Vector Quad Gen Difference Mode CircuitImplementation Simple Difference Amp.

6.2. Second Implementation

FIG. 30 illustrates Vector Quad Gen Block Diagrams/ABM SimulationCircuits.

FIG. 31 illustrates Vector Quad Gen Simplified Block Diagram—SimmerApproach.

FIG. 32 illustrates Vector Quad Gen Block Diagram/Transfer Functions.φ_(E1), φ_(E2), and φ_(E#) are phase errors associated with the 0/180,difference/90, and summer blocks, respectively. G_(x) are the gains ofeach block.

FIG. 33 illustrates Vector Quad Gen Phasor Diagrams.

According to an embodiment of the present invention, quadraturecomponent computation (referring to FIG. 32) is achieved according tothe following equations:S ₀ =[A ₁<0+(0.5)(A ₁<0−A ₂<180φ_(E1))(G ₁<90+φ_(E2))]G ₂<φ_(E3)S ₉₀ =[A ₂<180+φ_(E1)+(0.5)(A ₁<0−A ₂<180+φ_(E1))(G ₁<90φ_(E2))]G₂<φ_(E3)

It is apparent in this representation that the gain term G₂<φ_(E3)affects each phasor equally and does not influence the relativemagnitude or phase difference. Simplifying the above equations:S _(0a) =A ₁<0+(0.5)(A ₁ G ₁<90+φ_(E2))+(0.5)(A ₂ G ₁<90+φ_(E1)+φ_(E2))S _(90a) =−A ₂<φ_(E1)+(0.5)(A ₁ G ₁<90+φ_(E2))+(0.5)(A ₂ G₁<90+φ_(E1)+φ_(E2))

Accordingly, computing vector errors as a function of the individualblock phase errors and gains, with S₀ as reference, we can write:$\begin{matrix}{\quad{S_{90{a{({IDEAL})}}} = {\left( S_{\quad{0\quad a}} \right)\left( {1{\angle 90}} \right)}}} \\{= {{A_{1}{\angle 90}} - {(0.5)\left( {A_{1}G_{1}{\angle\phi}_{E\quad 2}} \right)} - {(0.5)\left( {{A_{2}G_{1}{\angle\phi}_{E\quad 1}} + \phi_{E\quad 2}} \right)}}}\end{matrix}$   S_(90a(ERROR)) = S_(90a(IDEAL)) − S_(90a)S_(90a(ERROR)) = +j  A₁ − (0.5)[A₁G₁cos (ϕ_(E  2))] − j(0.5)[A₁G₁sin (ϕ_(E  2))] − (0.5)[A₂G₁cos (ϕ_(E  1) + ϕ_(E  2))] − j(0.5)[A₂G₁sin (ϕ_(E  1) + ϕ_(E  2))] + A₂cos (ϕ_(E  1)) + j  A₂sin (ϕ_(E  1)) + (0.5)[A₁G₁sin (ϕ_(E  2))] − j(0.5)[A₁G₁cos (ϕ_(E  2))] + (0.5)[A₂G₁sin (ϕ_(E  1) + ϕ_(E  2))] − j(0.5)[A₂G₁cos (ϕ_(E  1) + ϕ_(E  2))]

If the 90° phase shift error φ_(E2) is small, the above equationssimplify to:S_(90a(ERROR)) = +j  A₁ − (0.5)(A₁G₁) − (0.5)A₂G₁cos (ϕ_(E  1)) − j(0.5)A₂G₁sin (ϕ_(E  1)) + A₂cos (ϕ_(E  1)) + j  A₂sin (ϕ_(E  1)) − j(0.5)A₁G₁ + (0.5)A₂G₁sin (ϕ_(E  1)) − j(0.5)A₂G₁cos (ϕ_(E  1))S_(90a(ERROR)) = −(0.5)(A₁G₁) + A₂cos (ϕ_(E  1)) + (0.5)A₂G₁[sin (ϕ_(E  1)) − cos (ϕ_(E  1))] + j  A₁(1 − 0.5G₁) + j  A₂sin (ϕ_(E  1)) − j(0.5)A₂G₁[sin (ϕ_(E  1)) + cos (ϕ_(E  1))]

Further, if G₁ is approximately unity, we then have:

In contrast to the vector error, the quadrature error or phase error canbe defined in terms of the relative phase of each of the quadraturecomponents compared to the ideal 90°:  ϕ_(90(ERROR)) = ϕ₉₀ − ϕ₀ − 90^(∘)S_(0a) = A₁(0.5)[A₁G₁sin (ϕ_(E  2))] + j(0.5)[A₁G₁cos (ϕ_(E  2))] − (0.5)[A₂G₁sin (ϕ_(E  1) + ϕ_(E  2))] + j(0.5)[A₂G₁cos (ϕ_(E  1) + ϕ_(E  2))]$\quad{{\tan\left( \phi_{0a} \right)} = \frac{0.5{G_{1}\left\lbrack {{A_{1}{\cos\left( \phi_{E\quad 2} \right)}} + {A_{2}{\cos\left( {\phi_{E\quad 1} + \phi_{E\quad 2}} \right)}}} \right\rbrack}}{A_{1} - {0.5{G_{1}\left\lbrack {{A_{1}{\sin\left( \phi_{E\quad 2} \right)}} + {A_{2}{\sin\left( {\phi_{E\quad 1} + \phi_{E\quad 2}} \right)}}} \right\rbrack}}}}$If  ϕ_(E  2)  is  small:$\quad{{\tan\left( \phi_{0a} \right)} \cong \frac{0.5{G_{1}\left\lbrack {A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}} \right\rbrack}}{A_{1} - {0.5{G_{1}\left\lbrack {A_{2}{\sin\left( \phi_{E\quad 1} \right)}} \right\rbrack}}}}$$\quad{{\tan\left( \phi_{90a} \right)} \cong \frac{{{- A_{2}}{\sin\left( \phi_{E\quad 1} \right)}} + {0.5{G_{1}\left\lbrack {A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}} \right\rbrack}}}{{{- A_{2}}{\cos\left( \phi_{E\quad 1} \right)}} - {0.5{G_{1}\left\lbrack {A_{2}{\sin\left( \phi_{E\quad 1} \right)}} \right\rbrack}}}}$

It should be noted that the gain term G₁ is dependent on input phasemismatch (φ_(E1)) with this particular R-C implementation. A 10° φ_(E1)will result in approximately 10% change in G₁. This dependenceapproximately doubles the sensitivity of quadrature phase error to 0/180phase mismatch.

Defining phase error as:φ_(90a(ERROR))=φ_(90a)−φ_(0a)−90°it can be empirically shown that|φ_(90a(ERROR))|≅|φ_(E1)|0/180° input phase mismatch (φ_(E1)) produces a quadrature phase errorof similar magnitude.

FIG. 34 illustrates Vector Quad Gen Simulations Schematics.

FIG. 35 further illustrates Vector Quad Gen Simulation Schematics(cont).

FIG. 36 illustrates Vector Quad Gen Simulation Results.

FIG. 37 further illustrates Vector Quad Gen Simulation Results.

FIG. 38 further illustrates Vector Quad Gen Simulation Results.

FIG. 39 further illustrates Vector Quad Gen Simulation Results.

FIG. 40 further illustrates Vector Quad Gen Simulation Results.

FIG. 41 further illustrates Vector Quad Gen Simulation Results.

FIG. 42 further illustrates Vector Quad Gen Simulation Results.

FIG. 43 further illustrates Vector Quad Gen Simulation Results.

FIG. 44 further illustrates Vector Quad Gen Simulation Results.

6.3. Third Implementation

FIG. 45 illustrates a Tuned Differential R-C Quad Gen Block Diagram.

FIG. 46 illustrates Tuned Differential R-C Quad Gen Block TransferFunctions. φ_(E1).φ_(E2), and φ_(E3) are phase errors associated withthe 1^(st) 1/180, 0/90, and 2^(nd) 0/180 pause split blocksrespectively. G_(x) are the gains of each block.

FIG. 47 illustrates Tuned Differential R-C Quad Gen Phasor Diagrams.

According to an embodiment of the present invention, Tuned Different R-CQuad Gen Block Transfer Functions can be written as:S ₀=[(A ₁<0)(G ₁<0)−(A ₂<180+φ_(E1))(G ₁<0)]G ₃<0S ₁₈₀=[(A ₁<0)(G ₁<0)−(A ₂<180+φ_(E1))(G ₁<0)]G ₄<180+φ_(E3)S ₂₇₀=[(A ₂<180φ_(E1))(G ₂<90+φ_(E2))−(A ₁<0)(G ₂<90φ_(E2))]G ₃<0S ₉₀=[(A ₂<180+φ_(E1))(G ₂<90+φ_(E2))−(A ₁<0)(G ₂<90+φ_(E2))]G₄<180+φ_(E3)

The above equations can be written in simplified form, as follows:S ₀ =A ₁ G ₁ G ₃<0+A ₂ G ₁ G ₃<φ_(E1)S ₁₈₀ =−A ₁ G ₁ G ₄<φ_(E3) −A ₂ G ₁ G ₄<φ_(E1)+φ_(E3)S ₂₇₀ =−A ₂ G ₂ G ₃90+φ_(E1)+φ_(E2) −A ₁ G ₂ G ₃<90+φ_(E2)S ₉₀ =A ₂ G ₂ G ₄<90+φ_(E1)+φ_(E2)+φ_(E3) +A ₁ G ₂ G ₄<90+φ_(E2)+φ_(E3)

Accordingly, vector error can be computed as a function of the variousindividual block phase errors and gains with So as reference, asfollows:   S_(180(IDEAL)) = −S₀ = −A₁G₁G₃∠0 − A₂G₁G₃∠ϕ_(E  1)  S_(180(ERROR)) = S_(180(IDEAL)) − S₁₈₀S_(180(ERROR)) = −A₁G₁G₃ − A₂G₁G₃cos (ϕ_(E  1)) − j  A₂G₁G₃sin (ϕ_(E  1)) + A₁G₁G₄cos (ϕ_(E  3)) + j  A₁G₁G₄sin (ϕ_(E  3)) + A₂G₁G₄cos (ϕ_(E  1) + ϕ_(E  3)) + j  A₂G₁G₄sin (ϕ_(E  1) + ϕ_(E  3))Similarly: $\begin{matrix}{S_{90{({ERROR})}} = {{{+ {jA}_{1}}G_{1}G_{3}} - {A_{2}G_{1}G_{3}{\sin\left( \phi_{E\quad 1} \right)}} + {{jA}_{2}G_{1}G_{3}{\cos\left( \phi_{E\quad 1} \right)}} +}} \\{{A_{2}G_{2}G_{4}{\sin\left( {\phi_{E\quad 1} + \phi_{E\quad 2} + \phi_{E\quad 3}} \right)}} -} \\{{{jA}_{2}G_{2}G_{4}{\cos\left( {\phi_{E\quad 1} + \phi_{E\quad 2} + \phi_{E\quad 3}} \right)}} +} \\{{A_{1}G_{2}G_{4}{\sin\left( {\phi_{E\quad 2} + \phi_{E\quad 3}} \right)}} - {{jA}_{1}G_{2}G_{4}{\cos\left( {\phi_{E\quad 2} + \phi_{E\quad 3}} \right)}}}\end{matrix}$ $\begin{matrix}{S_{270{({ERROR})}} = {{{- {jA}_{1}}G_{1}G_{3}} + {A_{2}G_{1}G_{3}{\sin\left( \phi_{E\quad 1} \right)}} - {{jA}_{2}G_{1}G_{3}{\cos\left( \phi_{E\quad 1} \right)}} -}} \\{{A_{2}G_{2}G_{3}{\sin\left( {\phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} + {{jA}_{2}G_{2}G_{3}{\cos\left( {\phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} -} \\{{A_{1}G_{2}G_{3}{\sin\left( \phi_{E\quad 2} \right)}} + {{jA}_{1}G_{2}G_{3}{\cos\left( \phi_{E\quad 2} \right)}}}\end{matrix}$As an example, further examining S_(180(ERROR)):S_(180(ERROR)) = −A₁G₁G₃ − A₂G₁G₃cos (ϕ_(E  1)) − jA₂G₁G₃sin (ϕ_(E  1)) + A₁G₁G₄cos (ϕ_(E  3)) + j  A₁G₁G₄sin (ϕ_(E  3)) + A₂G₁G₄[cos (ϕ_(E  1))cos (ϕ_(E  3)) − sin (ϕ_(E  1))sin (ϕ_(E  3))] + jA₂G₁G₄[sin (ϕ_(E  1))cos (ϕ_(E  3)) − cos (ϕ_(E  1))sin (ϕ_(E  3))]  

The input phase error, φ_(E1), may be significant. φ_(E2) and φ_(E3)will be determined by the matching of on chip components. Assumingφ_(E2) and φ_(E3) are small: $\begin{matrix}{S_{180{({ERROR})}} \cong {{{- A_{1}}G_{1}G_{3}} - {A_{2}G_{1}G_{3}{\cos\left( \phi_{E\quad 1} \right)}} - {{jA}_{2}G_{1}G_{3}{\sin\left( \phi_{E\quad 1} \right)}} +}} \\{{A_{1}G_{1}G_{4}} + {{jA}_{1}G_{1}{G_{4}\left( \phi_{E\quad 3} \right)}} +} \\{{A_{2}G_{1}{G_{4}\left\lbrack {{\cos\left( \phi_{E\quad 1} \right)} - {{\sin\left( \phi_{E\quad 1} \right)}\left( \phi_{E\quad 3} \right)}} \right\rbrack}} +} \\{{jA}_{2}G_{1}{G_{4}\left\lbrack {{\sin\left( \phi_{E\quad 1} \right)} - {{\cos\left( \phi_{E\quad 1} \right)}\left( \phi_{E\quad 3} \right)}} \right\rbrack}}\end{matrix}$ $\begin{matrix}{S_{180{({ERROR})}} \cong {{A_{1}{G_{1}\left( {G_{4} - G_{3}} \right)}} + {A_{2}{G_{1}\left\lbrack {{G_{4}{\cos\left( \phi_{E\quad 1} \right)}} - {G_{3}{\cos\left( \phi_{E\quad 1} \right)}}} \right\rbrack}} +}} \\{{jA}_{2}{G_{1}\left\lbrack {{G_{4}{\sin\left( \phi_{E\quad 1} \right)}} - {G_{3}{\sin\left( \phi_{E\quad 1} \right)}}} \right\rbrack}}\end{matrix}$

The gain associated with the two 0/180 splitter outputs, G₃ and G₄, arethe differential outputs of a soft limiting tuned amplifier fed by ashared tail current. Amplitude matching will be very good. Accordingly:$\begin{matrix}{S_{180{({ERROR})}} \cong {{A_{1}{G_{1}\left( {G_{4} - G_{3}} \right)}} + {A_{2}G_{1}{\cos\left( \phi_{E\quad 1} \right)}\left( {G_{4} - G_{3}} \right)} +}} \\{{jA}_{2}G_{1}{\sin\left( \phi_{E\quad 1} \right)}\left( {G_{4} - G_{3}} \right)} \\{\cong 0}\end{matrix}$

Regardless of the input amplitude matching (A₁, A₂) or input phase error(φ_(E1)), and assuming good 0/180 gain matching (G₃=G₄), the errorvector, S_(180(ERROR)), will be small.

In contrast to the vector error, the quadrature error or phase error canbe defined in terms of the relative phase or each of the quadraturecomponents compared to ideal. For example:ϕ_(180(ERROR)) = ϕ₁₈₀ − ϕ₀ − 180S₀ = A₁G₁G₃ + A₂G₁G₃cos (ϕ_(E  1)) − jA₂G₁G₃sin (ϕ_(E  1))${\tan\left( \phi_{0} \right)} = {\frac{A_{2}G_{1}G_{3}{\sin\left( \phi_{E\quad 1} \right)}}{{A_{1}G_{1}G_{3}} + {A_{2}G_{1}G_{3}{\cos\left( \phi_{E\quad 1} \right)}}} = \frac{A_{2}{\sin\left( \phi_{E\quad 1} \right)}}{A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}}}$

Similarly: $\begin{matrix}{S_{180} = {{{- A_{1}}G_{1}G_{4}{\cos\left( \phi_{E\quad 3} \right)}} - {{jA}_{1}G_{1}G_{4}{\sin\left( \phi_{E\quad 3} \right)}} -}} \\{{A_{2}G_{1}G_{4}{\cos\left( {\phi_{E\quad 1} + \phi_{E\quad 3}} \right)}} - {{jA}_{2}G_{1}G_{4}{\sin\left( {\phi_{E\quad 1} + \phi_{E\quad 3}} \right)}}}\end{matrix}$${\tan\left( \phi_{180} \right)} = \frac{{- G_{1}}{G_{4}\left( {{A_{1}{\sin\left( \phi_{E\quad 3} \right)}} + {A_{2}{\sin\left( {\phi_{E\quad 1} + \phi_{E\quad 3}} \right)}}} \right)}}{{- G_{1}}{G_{4}\left( {{A_{1}{\cos\left( \phi_{E\quad 3} \right)}} + {A_{2}{\cos\left( {\phi_{E\quad 1} + \phi_{E\quad 3}} \right)}}} \right)}}$

If φ_(E1) is small, the above equations can be re-written as:${\tan\left( \phi_{180} \right)} = \frac{{- A_{2}}{\sin\left( \phi_{E\quad 1} \right)}}{- \left\lbrack {A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}} \right\rbrack}$${\tan\left( \phi_{0} \right)} = \frac{A_{2}{\sin\left( \phi_{E\quad 1} \right)}}{A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}}$

It is now observed that φ₁₈₀=φ₀+180 and that φ_(180(ERROR))=0,regardless of input amplitude (A₁, A₂) or phase mismatch (φ_(E1)).

Similarly, we can examine the 270° quadrature component:S₂₇₀ = −A₂G₂G₃∠  90 + ϕ_(E  1) + ϕ_(E  2) − A₁G₂G₃∠  90 + ϕ_(E  2)ϕ_(270(ERROR)) = ϕ₂₇₀ − ϕ₀ − 270 $\begin{matrix}{S_{270} = {{{- A_{2}}G_{2}G_{3}{\cos\left( {90 + \phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} -}} \\{{{jA}_{2}G_{2}G_{3}{\sin\left( {90 + \phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} -} \\{{A_{1}G_{2}G_{3}{\cos\left( {90 + \phi_{E\quad 2}} \right)}} - {{jA}_{1}G_{2}G_{3}{\sin\left( {90 + \phi_{E\quad 2}} \right)}}}\end{matrix}$${\tan\left( \phi_{270} \right)} = \frac{{- G_{2}}{G_{3}\left\lbrack {{A_{2}{\sin\left( {90 + \phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} + {A_{1}{\sin\left( {90 + \phi_{E\quad 2}} \right)}}} \right\rbrack}}{{- G_{2}}{G_{3}\left\lbrack {{A_{2}{\cos\left( {90 + \phi_{E\quad 1} + \phi_{E\quad 2}} \right)}} + {A_{1}{\cos\left( {90 + \phi_{E\quad 2}} \right)}}} \right\rbrack}}$

If φ_(E2) is small, the above equations can be re-written as:${\tan\left( \phi_{270} \right)} = {\frac{- \left\lbrack {{A_{2}{\sin\left( {90 + \phi_{E\quad 1}} \right)}} + A_{1}} \right\rbrack}{{- A_{2}}{\cos\left( {90 + \phi_{E\quad 1}} \right)}} = \frac{- \left\lbrack {{A_{2}{\cos\left( \phi_{E\quad 1} \right)}} + A_{1}} \right\rbrack}{A_{2}{\sin\left( \phi_{E\quad 1} \right)}}}$${\tan\left( \phi_{0} \right)} = \frac{A_{2}{\sin\left( \phi_{E\quad 1} \right)}}{A_{1} + {A_{2}{\cos\left( \phi_{E\quad 1} \right)}}}$ϕ_(270(ERROR)) = ϕ₂₇₀ − ϕ₀ − 270 = tan¹(−X/Y) − tan⁻¹(Y/X) − 270ϕ_(270(ERROR)) = 0

regardless of input amplitude (A₁, A₂) or phase mismatch (φ_(E1)).

On the other hand, if φ_(E1) is ignored and φ_(E2) contributions tophase error are evaluated:${\tan\left( \phi_{270} \right)} = {\frac{- \left\lbrack {{A_{2}{\cos\left( \phi_{E\quad 2} \right)}} + {A_{1}{\cos\left( \phi_{E\quad 2} \right)}}} \right\rbrack}{{A_{2}{\sin\left( \phi_{E\quad 2} \right)}} + {A_{1}{\sin\left( \phi_{E\quad 2} \right)}}} = {- {\tan\left( \phi_{E\quad 2} \right)}}}$ϕ₂₇₀ = −ϕ_(E  2)

Since φ₀ is not a function of φ_(E2), |φ_(270(ERROR))|=|φ_(E2), asdetermined by the pass RC phase splitter.

FIG. 48 illustrates Tuned Differential R-C Quad Gen Differential CascodeCircuit Implementation.

FIG. 49 illustrates Tuned Differential R-C Quad Gen SimulationResults—Amplitude and Phase.

FIG. 50 illustrates Tuned Differential R-C Quad Gen Simulation Results.

FIG. 51 illustrates Tuned Differential R-C Quad Gen Simulation Results.

FIG. 52 illustrates Tuned Differential R-C Quad Gen Simulation Results(M3M4 OCT Inductor).

FIG. 53 illustrates Tuned Differential R-C Quad Gen Simulation Results.

FIG. 54 illustrates Differential Tuned Amp Transfer Gain Performs SoftLimiter Function.

IV. Conclusion

The present invention has been described above with the aid offunctional building blocks illustrating the performance of functions andrelationships thereof. At least some of the boundaries of thesefunctional building blocks have been arbitrarily defined herein for theconvenience of the description. Alternate boundaries can be defined solong as the specified functions and relationships thereof areappropriately performed. Any such alternate boundaries are thus withinthe scope and spirit of the claimed invention. One skilled in the artwill recognize that these functional building blocks can be implementedby discrete components, application specific integrated circuits,processors executing appropriate software and the like and combinationsthereof.

It is to be appreciated that the Detailed Description section, and notthe Summary and Abstract sections, is intended to be used to interpretthe claims. The Summary and Abstract sections can set forth one or more,but not all exemplary embodiments of the present invention ascontemplated by the inventor(s), and thus, are not intended to limit thepresent invention and the appended claims in any way.

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. It will be apparent to persons skilledin the relevant art that various changes in form and detail can be madetherein without departing from the spirit and scope of the invention.Thus, the breadth and scope of the present invention should not belimited by any of the above-described exemplary embodiments, but shouldbe defined only in accordance with the following claims and theirequivalents.

1. A method of generating orthogonal vectors, comprising: (a) generating first and second approximately orthogonal vector signals, {right arrow over (V)}₁ and {right arrow over (V)}₄, from a clock signal; (b) combining the first and second vector signals {right arrow over (V)}₁ and {right arrow over (V)}₄, thereby generating a vector signal I; and (c) combining an inverted version, −{right arrow over (V)}₁, of the first vector signal {right arrow over (V)}₁ with the second vector signal {right arrow over (V)}₄, thereby generating a vector signal {right arrow over (Q)}; whereby the vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially orthogonal to one another.
 2. The method of claim 1, wherein step (c) comprises: receiving the inverted version −{right arrow over (V)}₁; and summing the inverted version −{right arrow over (V)}₁ with the second vector signal {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)} substantially orthogonal to the vector signal {right arrow over (I)}.
 3. The method of claim 1, wherein step (c) comprises: generating the inverted version −{right arrow over (V)}₁; and summing the inverted version −{right arrow over (V)}₁ with the second vector signal {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)} substantially orthogonal to the vector signal {right arrow over (I)}.
 4. The method of claim 1, wherein step (c) comprises subtracting the first signal vector {right arrow over (V)}₁ from the second signal vector {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)}, whereby phase errors between vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially cancelled.
 5. The method of claim 3, wherein the generating an inverted version −{right arrow over (V)}₁ comprises inverting the vector signal {right arrow over (V)}₁, whereby −{right arrow over (V)}₁ is equal in amplitude and opposite in phase.
 6. The method of claim 1, wherein step (c) comprises one or more linear operations.
 7. The method of claim 1, wherein step (a) comprises: receiving a clock signal; generating the vector signal {right arrow over (V)}₁ and an inverted version, −{right arrow over (V)}₁, of the vector signal {right arrow over (V)}₁ from the clock signal; and generating vector signal {right arrow over (V)}₄ from vector signals {right arrow over (V)}₁ and −{right arrow over (V)}₁; whereby vector signals {right arrow over (V)}₁ and {right arrow over (V)}₄ are approximately orthogonal.
 8. The method of claim 7, wherein the generating vector signal {right arrow over (V)}₄ from vector signals {right arrow over (V)}₁ and −{right arrow over (V)}₁ comprises phase shifting {right arrow over (V)}₁ and/or −{right arrow over (V)}₁.
 9. The method of claim 7, wherein the generating vector signal {right arrow over (V)}₄ from vector signals {right arrow over (V)}₁ and −{right arrow over (V)}₁ comprises: subtracting −{right arrow over (V)}₁ from {right arrow over (V)}₁, thereby generating the vector signal {right arrow over (V)}₄; and phase shifting vector signal {right arrow over (V)}₄.
 10. The method of claim 7, wherein the generating an inverted version −{right arrow over (V)}₁ comprises phase shifting the clock signal 180°, thereby generating −{right arrow over (V)}₁.
 11. A method of generating orthogonal vectors, comprising: (a) generating a first vector signal {right arrow over (V)}₁ and an inverted version, −{right arrow over (V)}₁, of the first vector signal {right arrow over (V)}₁, from a clock signal; (b) combining the first vector signal {right arrow over (V)}₁ and an approximately orthogonal third vector signal {right arrow over (V)}₄, thereby generating a vector signal I; and (c) combining the inverted version −{right arrow over (V)}₁ with the vector signal {right arrow over (V)}₄, thereby generating a vector signal {right arrow over (Q)}; whereby the vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially orthogonal to one another.
 12. The method of claim 11, wherein step (b) comprises receiving the vector signal {right arrow over (V)}₄.
 13. The method of claim 11, wherein step (b) comprises generating the vector signal {right arrow over (V)}₄.
 14. The method of claim 13, wherein the generating vector signal {right arrow over (V)}₄ comprises phase shifting {right arrow over (V)}₁ and/or −{right arrow over (V)}₁.
 15. The method of claim 13, wherein the generating vector signal {right arrow over (V)}₄ comprises: subtracting −{right arrow over (V)}₁ from {right arrow over (V)}₁, thereby generating the vector signal {right arrow over (V)}₄; and phase shifting vector signal {right arrow over (V)}₄.
 16. The method of claim 11, wherein step (c) comprises summing the inverted version −{right arrow over (V)}₁ with the third vector signal {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)} substantially orthogonal to the vector signal {right arrow over (I)}.
 17. The method of claim 11, wherein step (c) comprises subtracting the signal vector {right arrow over (V)}₁ from the signal vector {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)}, whereby phase errors between vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially cancelled.
 18. The method of claim 11, wherein step (c) comprises one or more linear operations.
 19. The method of claim 11, wherein step (a) comprises: subtracting a second clock signal VB from a first clock signal VA, thereby generating a first vector signal {right arrow over (V)}₁; and subtracting the first clock signal VA from the second clock signal VB, thereby generating an inverted version, −{right arrow over (V)}₁, of the first vector signal {right arrow over (V)}₁; whereby −{right arrow over (V)}₁ is equal in amplitude and opposite in phase to {right arrow over (V)}₁.
 20. A method of generating orthogonal vectors, comprising: (a) receiving first and second approximately orthogonal vector signals, {right arrow over (V)}₁ and {right arrow over (V)}₄; (b) combining the first and second vector signals {right arrow over (V)}₁ and {right arrow over (V)}₄, thereby generating a vector signal {right arrow over (I)}; and (c) combining an inverted version, −{right arrow over (V)}₁, of the first vector signal {right arrow over (V)}₁ with the second vector signal {right arrow over (V)}₄, thereby generating a vector signal {right arrow over (Q)}; whereby the vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially orthogonal to one another.
 21. The method of claim 20, wherein step (c) comprises: receiving the inverted version −{right arrow over (V)}₁; and summing the inverted version −{right arrow over (V)}_(‘)with the second vector signal {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)} substantially orthogonal to the vector signal {right arrow over (I)}.
 22. The method of claim 20, wherein step (c) comprises: generating the inverted version −{right arrow over (V)}₁; and summing the inverted version −{right arrow over (V)}₁ with the second vector signal {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)} substantially orthogonal to the vector signal {right arrow over (I)}.
 23. The method of claim 20, wherein step (c) comprises subtracting the first signal vector {right arrow over (V)}₁ from the second signal vector {right arrow over (V)}₄, thereby generating the vector signal {right arrow over (Q)}, whereby phase errors between vector signals {right arrow over (I)} and {right arrow over (Q)} are substantially cancelled.
 24. The method of claim 22, wherein the generating an inverted version −{right arrow over (V)}₁ comprises inverting the vector signal {right arrow over (V)}₁, whereby −{right arrow over (V)}₁ is equal in amplitude and opposite in phase.
 25. The method of claim 20, wherein step (c) comprises one or more linear operations.
 26. A system for generating substantially orthogonal signals {right arrow over (I)} and {right arrow over (Q)} from approximately orthogonal signals {right arrow over (V)}₁, {right arrow over (V)}₄, and an inverted version of {right arrow over (V)}₁, −{right arrow over (V)}₁ comprising: a first combiner configured to receive {right arrow over (V)}₁ and {right arrow over (V)}₄ and to generate the signal {right arrow over (I)}; a second combiner configured to receive −{right arrow over (V)}₁ and {right arrow over (V)}₄ and to generate the signal {right arrow over (Q)}; whereby the signals {right arrow over (I)} and {right arrow over (Q)} are substantially orthogonal to one another.
 27. The system of claim 26, wherein the system comprises: an approximate orthogonal signal generator, configured to receive a clock signal and output the approximately orthogonal signals {right arrow over (V)}₁ and {right arrow over (V)}₄, coupled to the first and second combiners; an inverter configured to receive {right arrow over (V)}₁ and output −{right arrow over (V)}₁ coupled to the signal generator and the second combiner.
 28. The system of claim 26, wherein the system comprises: a signal generator, configured to receive a clock signal and output vector signal {right arrow over (V)}₁ and an inverted version of {right arrow over (V)}₁, −{right arrow over (V)}₁; a phase shifter coupled to the signal generator configured to receive {right arrow over (V)}₁ and output {right arrow over (V)}₄ approximately orthogonal to {right arrow over (V)}₁; the signal generator coupled to the first combiner and second combiners; the phase shifter coupled to the first and second combiners.
 29. The system of claim 27, wherein the approximate orthogonal signal generator comprises: a first phase shifter configured to receive a clock signal, the first phase shifter coupled to a first, second, and third linear operator, a second phase shifter coupled to the first, second, and third linear operators, the first linear operator coupled to the second and third linear operators, and the second and third linear operators coupled to each other; wherein the second and third linear operators each output vector signals {right arrow over (V)}₁ and {right arrow over (V)}₄ approximately orthogonal to one another.
 30. The system of claim 29, wherein the first, second and third linear operators are closely matched differential operators.
 31. The system of claim 29, wherein the first, second and third linear operators are closely matched summing operators.
 32. The system of claim 26, wherein the second combiner further comprises a differencer.
 33. The system of claim 27, wherein the second combiner further comprises a differencer.
 34. The system of claim 28, wherein the second combiner further comprises a differencer.
 35. The system of claim 28, wherein the signal generator comprises: a first differential node coupled to first and second terminals of a clock signal generator; the first differential node configured to receive the clock signal and output vector signal {right arrow over (V)}₁; a second differential node cross-coupled to the first and second terminals of the clock signal generator; the second differential node configured to receive the clock signal and output inverted version −{right arrow over (V)}₁.
 36. A method of generating orthogonal vectors, comprising: (a) receiving a clock signal {right arrow over (S)}₁; (b) phase shifting the signal {right arrow over (S)}₁ 180°, thereby generating a signal {right arrow over (S)}₂; (c) phase shifting the signal {right arrow over (S)}₁ 90°, thereby generating a signal {right arrow over (S)}₃; (d) phase shifting the signal {right arrow over (S)}₂ 90°, thereby generating a signal {right arrow over (S)}₄; (e) subtracting the signal {right arrow over (S)}₂ from the signal {right arrow over (S)}₁, thereby generating a signal {right arrow over (V)}₁; (f) phase shifting the signal {right arrow over (V)}₁ 180°, thereby generating a signal {right arrow over (V)}₃; (g) subtracting {right arrow over (S)}₃ from {right arrow over (S)}₄, thereby generating a signal {right arrow over (V)}₄; and (h) phase shifting the signal {right arrow over (V)}₄ 180°, thereby generating a signal {right arrow over (V)}₂; whereby {right arrow over (V)}₂ is approximately orthogonal to {right arrow over (V)}₁, {right arrow over (V)}₃ is approximately opposite in phase to {right arrow over (V)}₁, and {right arrow over (V)}₄ is approximately orthogonal to {right arrow over (V)}₁.
 37. A system for generating orthogonal vector signals, comprising: a first phase shifter configured to receive a clock signal; the first phase shifter coupled to second and third phase shifters, the second and third phase shifters cross-coupled to a first and second linear operator, the first and second phase linear operators coupled to fourth and fifth phase shifters, wherein the fourth and fifth phase shifters output vector signals which are approximately orthogonal to one another.
 38. The system of claim 37, wherein the first and second linear operators are closely matched differential operators.
 39. The system of claim 37, wherein the first and second linear operators are closely matched summing operators. 